Families of permutations with a forbidden intersection

A family of permutations is said to be ‘t-intersecting’ if any two permutations in the family agree on at least t points. It is said to be (t-1)-intersection-free if no two permutations in the family agree on exactly t-1 points. Deza and Frankl conjectured in 1977 that a
t-intersecting family of permutations in S_n can be no larger than a coset of the stabiliser of t points, provided n is large enough depending on t; this was proved by the speaker and independently by Friedgut and Pilpel in 2008. We give a new proof of a stronger statement: namely, that a (t-1)-intersection-free family of permutations in S_n can be no larger than a coset of the stabiliser of t points, provided n is large enough. This can be seen as an analogue for permutations of seminal results of Frankl and Furedi on families of k-element sets. Our proof is partly algebraic and
partly combinatorial; it is more ‘robust’ than the original proofs of the Deza-Frankl conjecture, using a combinatorial ‘quasirandomness’ argument to avoid many of the algebraic difficulties of the original proofs. Based on joint work with Noam Lifshitz (Bar Ilan University).